Derivation of Oil Flow in an Ancient Oil Lamp

In an ancient oil lamp, the flow of oil through the wick to the flame is primarily driven by capillary action and resisted by the viscosity of the oil. The balance between these two forces determines the height to which oil can rise in the wick and the rate of flow to the flame.

Key Variables

$ \gamma $ = Surface tension of the oil (N/m)
$ \eta $ = Viscosity of the oil (Pa·s)
$ \rho $ = Density of the oil (kg/m³)
$ r $ = Radius of the wick pores (m)
$ L $ = Length of the wick (m)
$ h $ = Height to which the oil rises in the wick due to capillary action (m)
$ g $ = Acceleration due to gravity (m/s²)

Step 1: Capillary Rise in the Wick

The height $ h $ to which the oil can rise due to capillary action in a cylindrical pore of the wick is given by the capillary rise formula:

$$ h = \frac{2 \gamma \cos \theta}{\rho g r} $$

where:

$ \theta $ is the contact angle between the oil and the wick material.
$ r $ is the radius of the capillary pores within the wick.

This height represents the maximum capillary rise, beyond which gravity counteracts further rise.

Step 2: Flow Rate of Oil Due to Viscosity

As oil rises through the wick, it encounters resistance due to viscosity. The flow rate $ Q $ (volume per time) through a cylindrical pore in the wick due to viscosity can be described by Poiseuille's law:

$$ Q = \frac{\Delta P \cdot \pi r^4}{8 \eta L} $$

where:

$ \Delta P $ is the pressure difference across the length of the wick $ L $.
$ \eta $ is the viscosity of the oil.

Step 3: Pressure Difference Due to Capillary Action

The pressure difference $ \Delta P $ generated by capillary action can be expressed as:

$$ \Delta P = \frac{2 \gamma \cos \theta}{r} $$

Substituting this into the flow rate equation:

$$ Q = \frac{\left( \frac{2 \gamma \cos \theta}{r} \right) \cdot \pi r^4}{8 \eta L} $$

Simplifying, we get:

$$ Q = \frac{\pi \gamma r^3 \cos \theta}{4 \eta L} $$

Step 4: Relation of Oil Weight to Wick and Flow

The weight $ W $ of the oil flowing per second (i.e., mass flow rate) is given by:

$$ W = Q \cdot \rho $$

Substituting the expression for $ Q $ from above, we get:

$$ W = \frac{\pi \gamma r^3 \cos \theta \cdot \rho}{4 \eta L} $$

Final Relation

The final relation governing the flow of oil in an ancient oil lamp, incorporating capillarity, viscosity, and wick length, is:

$$ W = \frac{\pi \gamma r^3 \cos \theta \cdot \rho}{4 \eta L} $$

Conclusion

This relationship shows that the weight of oil flowing in an ancient lamp wick depends on the surface tension $ \gamma $, pore radius $ r $, contact angle $ \theta $, oil density $ \rho $, viscosity $ \eta $, and length of the wick $ L $. Higher surface tension, larger pore radius, and lower viscosity allow for more efficient oil flow, sustaining the lamp flame longer.